It's pretty obvious that $\phi(0) = 0$ and $\phi(1) = 1$ so those are all set.
Now I want to show that $\phi(a+b) = \phi(a) + \phi(b)$
$(a+b)^p = a^p + b^p$
for all $a,b \in R$.
however it's not immediate how to do this. $R$ being commutative doesn't help here at all, and the characteristic $p$ applies to addition not exponentiation..